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Metamath Proof Explorer

Theorem xrltlen

Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015)

Ref Expression
Assertion xrltlen ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 xrlttri ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) )
2 ioran ⊢ ( ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ↔ ( ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴 ) )
3 2 biancomi ⊢ ( ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵 ) )
4 1 3 bitrdi ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵 ) ) )
5 xrlenlt ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
6 nesym ⊢ ( 𝐵 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐵 )
7 6 a1i ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐵 ) )
8 5 7 anbi12d ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵 ) ) )
9 4 8 bitr4d ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) ) )